Representations of the Weyl Algebra in Quantum Geometry

The Weyl algebra of continuous functions and exponentiated fluxes, introduced by Ashtekar, Lewandowski and others, in quantum geometry is studied. It is shown that, in the piecewise analytic category, every regular representation of having a cyclic and diffeomorphism invariant vector, is already unitarily equivalent to the fundamental representation. Additional assumptions concern the dimension of the underlying analytic manifold (at least three), the finite wide triangulizability of surfaces in it to be used for the fluxes and the naturality of the action of diffeomorphisms – but neither any domain properties of the represented Weyl operators nor the requirement that the diffeomorphisms act by pull-backs. For this, the general behaviour of C*-algebras generated by continuous functions and pull-backs of homeomorphisms, as well as the properties of stratified analytic diffeomorphisms are studied. Additionally, the paper includes also a short and direct proof of the irreducibility of .     

Unbounded Representations of q-Deformation of Cuntz Algebra

We study a deformation of the Cuntz–Toeplitz C ^*-algebra determined by the relations ${a_i^*a_i=1+q a_ia_i^*,\, a_i^*a_j=0}$ . We define its well-behaved unbounded *-representations and classify all irreducible ones up to unitary equivalence.     

The qs-Hermite Polynomial and the Representations of Heisenberg and Quantum Heisenberg Algebra

By introducing a pair of canonical conjugate two-parameter deformed operators D_qs, X_qs,we can naturally obtain the form of qs-analogous Taylor series for an arbitrary analytic function, and explicitly construct the realizations of Heisenberg and two-parameter deformed quantum Heisenberg algebra by means of the operators D_qs and X_qs, and it is shown that the qs-analogous Hermite polynomials are the representations of Heisenberg and the quantum Heisen berg algebra.     

Trefoil Symmetry V: Class Representations for the Minimal Clover Extension

We construct four different real representations of the minimal vector clover extension and their covariant derivatives. We determine their relation to the basic representation and initiate the exploration of constrained superfields.     

On the Classification of Admissible Representations of the Virasoro Algebra

For the Virasoro algebra, the problems of classifying bounded admissible representations, on the one hand, and finite-length extensions of highest weight modules, on the other hand, are wild: they are as complicated as the problem of classifying a pair of matrices. This follows from results by Martin and Piard and Feigin and Fuchs.     

The sl2 Loop Algebra Symmetry of the Six-Vertex Model at Roots of Unity

We demonstrate that the six vertex model (XXZ spin chain) with =(q+q^-1)/2 and q^2N=1 has an invariance under the loop algebra of sl₂ which produces a special set of degenerate eigenvalues. For =0 we compute the multiplicity of the degeneracies using Jordan–Wigner techniques.     

Letter: Self-Similar Collapse of Scalar Field with Plane Symmetry

Plane symmetric self-similar solutions to Einstein's four-dimensional theory of gravity are studied and all such solutions are given analytically in closed form. The local and global properties of these solutions are investigated and it is shown that some of the solutions can be interpreted as representing gravitational collapse of the scalar field. During the collapse, trapped surfaces are never developed. As a result, no black hole is formed. Although the collapse always ends with spacetime singularities, it is found that these singularities are spacelike and not naked.     

Representations of the Witt Algebra and Gl(n)-Opers

We exhibit a new link between certain representations of the Witt algebra and some Gl(n)-opers on the punctured disc. As applications, we discuss the connection with the KdV hierarchy and Virasoro constraints and how the Virasoro constraints of the so-called topological recursion fit in our approach.     

浅谈平面振子的阻尼运动

从动力学方程出发并结合Mathcad数值计算,研究了平面振子的阻尼运动,并分析了阻尼系数对平面振子运动轨迹的影响.     

The Cohomology of the Variational Bicomplex Invariant under the Symmetry Algebra of the Potential Kadomtsev-Petviashvili Equation

Abstract

The variational bicomplex of forms invariant under the symmetry algebra of the potential Kadomtsev-Petviashvili equation is described and the cohomology of the associated Euler-Lagrange complex is computed. The results are applied to a characterization problem of the Kadomtsev-Petviashvili equation by its symmetry algebra originally posed by David, Levi, and Winternitz.     

Irreducible Representations of Quantum sl(3) Enveloping Algebra

By referring to the construction method of finding representations of classical U(n) group, we developed a parallel method with which one can construct irreducible representations of quantum sl(3) enveloping algebra explicitly.     

Exact Solutions with Arbitrary Permutation Symmetry in a Three-Body Harmonic Oscillator

The exact solutions with arbitrary permutation symmetry to the threebody Schrödinger equation for the harmonic oscillator are presented completely.     

The Temperley-Lieb-Jones Algebra and Symmetry Group in Heisenberg Spin Models in One and Two Dimensions

We present a systematic method to construct the spin models of Heisenberg type in higher dimensions with nearesband non-nearest neighbour interactions. These models constructed in this way are of Temperley-Lieb-Jones (TLJ) algebraic structures and SU(2)-invariances. The TLJ algebra is generalized to adjusting the lattice spin models. The Hamiltonians of Heisenberg spin models in one dimension (including second nearest neighbour interactions) and in two-dimensional triangular lattice (with nearest interaction) are constructed explicitly. The hidden symmetries are shown to be the SU(2) group, and the terms in Ilamiltonian for different lattice cells are explicitly shown to be the representations of elements of (TLJ) algebra.     

Unitary Operator of su q (n)-Covariant Oscillator Algebra

The unitary operator of su _q (n)-covariant oscillator algebra is constructed and two types of q-coherent states are obtained explicitly.     

Non-homogeneous Polygonal Markov Fields in the Plane: Graphical Representations and Geometry of Higher Order Correlations

We consider polygonal Markov fields originally introduced by Arak and Surgailis (Probab. Theory Relat. Fields 80:543–579, 1989). Our attention is focused on fields with nodes of order two, which can be regarded as continuum ensembles of non-intersecting contours in the plane, sharing a number of features with the two-dimensional Ising model. We introduce non-homogeneous version of polygonal fields in anisotropic environment. For these fields we provide a class of new graphical constructions and random dynamics. These include a generalized dynamic representation, generalized and defective disagreement loop dynamics as well as a generalized contour birth and death dynamics. Next, we use these constructions as tools to obtain new exact results on the geometry of higher order correlations of polygonal Markov fields in their consistent regime. Research supported by the Polish Minister of Science and Higher Education grant N N201 385234 (2008-2010).     

On the Algebra of Symmetry of the Dirac Equation in Flat Space and De Sitter Space

The structure of the quadratic algebras of spinor symmetry operators for the Dirac equation is studied in a four-dimensional flat space and in the de Sitter space of arbitrary signature. The algebras are shown to be standard equivalent. Linear noncommutative subalgebras meeting the conditions of the noncommutative integrability theorem are found in these algebras.